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Theorem dihfval 31421
Description: Isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
Hypotheses
Ref Expression
dihval.b  |-  B  =  ( Base `  K
)
dihval.l  |-  .<_  =  ( le `  K )
dihval.j  |-  .\/  =  ( join `  K )
dihval.m  |-  ./\  =  ( meet `  K )
dihval.a  |-  A  =  ( Atoms `  K )
dihval.h  |-  H  =  ( LHyp `  K
)
dihval.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihval.d  |-  D  =  ( ( DIsoB `  K
) `  W )
dihval.c  |-  C  =  ( ( DIsoC `  K
) `  W )
dihval.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihval.s  |-  S  =  ( LSubSp `  U )
dihval.p  |-  .(+)  =  (
LSSum `  U )
Assertion
Ref Expression
dihfval  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) ) ) )
Distinct variable groups:    A, q    u, q, x, K    x, B    u, S    W, q, u, x
Allowed substitution hints:    A( x, u)    B( u, q)    C( x, u, q)    D( x, u, q)    .(+) ( x, u, q)    S( x, q)    U( x, u, q)    H( x, u, q)    I( x, u, q)    .\/ ( x, u, q)    .<_ ( x, u, q)    ./\ (
x, u, q)    V( x, u, q)

Proof of Theorem dihfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dihval.i . . 3  |-  I  =  ( ( DIsoH `  K
) `  W )
2 dihval.b . . . . 5  |-  B  =  ( Base `  K
)
3 dihval.l . . . . 5  |-  .<_  =  ( le `  K )
4 dihval.j . . . . 5  |-  .\/  =  ( join `  K )
5 dihval.m . . . . 5  |-  ./\  =  ( meet `  K )
6 dihval.a . . . . 5  |-  A  =  ( Atoms `  K )
7 dihval.h . . . . 5  |-  H  =  ( LHyp `  K
)
82, 3, 4, 5, 6, 7dihffval 31420 . . . 4  |-  ( K  e.  V  ->  ( DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) )
98fveq1d 5527 . . 3  |-  ( K  e.  V  ->  (
( DIsoH `  K ) `  W )  =  ( ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) `
 W ) )
101, 9syl5eq 2327 . 2  |-  ( K  e.  V  ->  I  =  ( ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w , 
( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) `
 W ) )
11 breq2 4027 . . . . 5  |-  ( w  =  W  ->  (
x  .<_  w  <->  x  .<_  W ) )
12 fveq2 5525 . . . . . . 7  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  ( ( DIsoB `  K ) `  W ) )
13 dihval.d . . . . . . 7  |-  D  =  ( ( DIsoB `  K
) `  W )
1412, 13syl6eqr 2333 . . . . . 6  |-  ( w  =  W  ->  (
( DIsoB `  K ) `  w )  =  D )
1514fveq1d 5527 . . . . 5  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  x )  =  ( D `  x ) )
16 fveq2 5525 . . . . . . . . 9  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  ( ( DVecH `  K ) `  W ) )
17 dihval.u . . . . . . . . 9  |-  U  =  ( ( DVecH `  K
) `  W )
1816, 17syl6eqr 2333 . . . . . . . 8  |-  ( w  =  W  ->  (
( DVecH `  K ) `  w )  =  U )
1918fveq2d 5529 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  (
LSubSp `  U ) )
20 dihval.s . . . . . . 7  |-  S  =  ( LSubSp `  U )
2119, 20syl6eqr 2333 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 ( ( DVecH `  K ) `  w
) )  =  S )
22 breq2 4027 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .<_  w  <->  q  .<_  W ) )
2322notbid 285 . . . . . . . . 9  |-  ( w  =  W  ->  ( -.  q  .<_  w  <->  -.  q  .<_  W ) )
24 oveq2 5866 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
x  ./\  w )  =  ( x  ./\  W ) )
2524oveq2d 5874 . . . . . . . . . 10  |-  ( w  =  W  ->  (
q  .\/  ( x  ./\  w ) )  =  ( q  .\/  (
x  ./\  W )
) )
2625eqeq1d 2291 . . . . . . . . 9  |-  ( w  =  W  ->  (
( q  .\/  (
x  ./\  w )
)  =  x  <->  ( q  .\/  ( x  ./\  W
) )  =  x ) )
2723, 26anbi12d 691 . . . . . . . 8  |-  ( w  =  W  ->  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  <-> 
( -.  q  .<_  W  /\  ( q  .\/  ( x  ./\  W ) )  =  x ) ) )
2818fveq2d 5529 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  (
LSSum `  U ) )
29 dihval.p . . . . . . . . . . 11  |-  .(+)  =  (
LSSum `  U )
3028, 29syl6eqr 2333 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSSum `  ( ( DVecH `  K ) `  w
) )  =  .(+)  )
31 fveq2 5525 . . . . . . . . . . . 12  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  ( ( DIsoC `  K ) `  W ) )
32 dihval.c . . . . . . . . . . . 12  |-  C  =  ( ( DIsoC `  K
) `  W )
3331, 32syl6eqr 2333 . . . . . . . . . . 11  |-  ( w  =  W  ->  (
( DIsoC `  K ) `  w )  =  C )
3433fveq1d 5527 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoC `  K
) `  w ) `  q )  =  ( C `  q ) )
3514, 24fveq12d 5531 . . . . . . . . . 10  |-  ( w  =  W  ->  (
( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) )  =  ( D `  ( x 
./\  W ) ) )
3630, 34, 35oveq123d 5879 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) )  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) )
3736eqeq2d 2294 . . . . . . . 8  |-  ( w  =  W  ->  (
u  =  ( ( ( ( DIsoC `  K
) `  w ) `  q ) ( LSSum `  ( ( DVecH `  K
) `  w )
) ( ( (
DIsoB `  K ) `  w ) `  (
x  ./\  w )
) )  <->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) )
3827, 37imbi12d 311 . . . . . . 7  |-  ( w  =  W  ->  (
( ( -.  q  .<_  w  /\  ( q 
.\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w ) `  q
) ( LSSum `  (
( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) )  <-> 
( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) )
3938ralbidv 2563 . . . . . 6  |-  ( w  =  W  ->  ( A. q  e.  A  ( ( -.  q  .<_  w  /\  ( q 
.\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w ) `  q
) ( LSSum `  (
( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) )  <->  A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) )
4021, 39riotaeqbidv 6307 . . . . 5  |-  ( w  =  W  ->  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K
) `  w )
) A. q  e.  A  ( ( -.  q  .<_  w  /\  ( q  .\/  (
x  ./\  w )
)  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) )  =  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) )
4111, 15, 40ifbieq12d 3587 . . . 4  |-  ( w  =  W  ->  if ( x  .<_  w ,  ( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) )  =  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) ) )
4241mpteq2dv 4107 . . 3  |-  ( w  =  W  ->  (
x  e.  B  |->  if ( x  .<_  w ,  ( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) )  =  ( x  e.  B  |->  if ( x  .<_  W ,  ( D `  x ) ,  (
iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) ) ) )
43 eqid 2283 . . 3  |-  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w , 
( ( ( DIsoB `  K ) `  w
) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x 
.<_  w ,  ( ( ( DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) )
44 fvex 5539 . . . . 5  |-  ( Base `  K )  e.  _V
452, 44eqeltri 2353 . . . 4  |-  B  e. 
_V
4645mptex 5746 . . 3  |-  ( x  e.  B  |->  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) ) )  e.  _V
4742, 43, 46fvmpt 5602 . 2  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( (
DIsoB `  K ) `  w ) `  x
) ,  ( iota_ u  e.  ( LSubSp `  (
( DVecH `  K ) `  w ) ) A. q  e.  A  (
( -.  q  .<_  w  /\  ( q  .\/  ( x  ./\  w ) )  =  x )  ->  u  =  ( ( ( ( DIsoC `  K ) `  w
) `  q )
( LSSum `  ( ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K
) `  w ) `  ( x  ./\  w
) ) ) ) ) ) ) ) `
 W )  =  ( x  e.  B  |->  if ( x  .<_  W ,  ( D `  x ) ,  (
iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
.\/  ( x  ./\  W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
 ( x  ./\  W ) ) ) ) ) ) ) )
4810, 47sylan9eq 2335 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  if ( x  .<_  W , 
( D `  x
) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q  .\/  (
x  ./\  W )
)  =  x )  ->  u  =  ( ( C `  q
)  .(+)  ( D `  ( x  ./\  W ) ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   LSSumclsm 14945   LSubSpclss 15689   Atomscatm 29453   LHypclh 30173   DVecHcdvh 31268   DIsoBcdib 31328   DIsoCcdic 31362   DIsoHcdih 31418
This theorem is referenced by:  dihval  31422  dihf11lem  31456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-dih 31419
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